What is the ruling of Islamic law on intervening to reconcile a brother and his wife, after she swore by Allah that her son would be paralyzed if she intervened in the reconciliation?

The correct approach is to reconcile between your brother and his wife, and to break your oath, because reconciling people (إصلاح ذات البين) is among the best deeds. You must then offer expiation for your oath. As for your oath concerning your son becoming paralyzed, there is no expiation for it, but such expressions should be avoided. The expiation for an oath is to feed ten poor people or clothe them, or to fast three days if you cannot find the means to do the former.